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We address the formation of massive stellar spheroids between redshifts $z=4$and 1 using a suite of AMR hydro-cosmological simulations. The spheroids formas bulges, and the spheroid mass growth is partly driven by violent discinstability (VDI) and partly by mergers. A kinematic decomposition to disc andspheroid yields that the mass fraction in the spheroid is between 50\% and 90\%and is roughly constant in time, consistent with a cosmological steady state ofVDI discs that are continuously fed from the cosmic web. The density profile ofthe spheroid is typically "classical", with a Sersic index $n = 4.5\pm 1$,independent of whether it grew by mergers or VDI and independent of thefeedback strength. The disc is characterized by $n=1.5\pm 0.5$, and the wholegalaxy by $n=3\pm 1$. The high-redshift spheroids are compact due to thedissipative inflow of gas and the high universal density. The stellar surfacedensity within the effective radius of each galaxy as it evolves remainsroughly constant in time after its first growth. For galaxies of a fixedstellar mass, the surface density is higher at higher redshifts.

Early formation of massive, compact, spheroidal galaxies with classical profiles by violent disc instability or mergers